# Properties

 Label 726.4.a.f Level $726$ Weight $4$ Character orbit 726.a Self dual yes Analytic conductor $42.835$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [726,4,Mod(1,726)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(726, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("726.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$726 = 2 \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 726.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$42.8353866642$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{5} - 6 q^{6} + 16 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10})$$ q + 2 * q^2 - 3 * q^3 + 4 * q^4 + 6 * q^5 - 6 * q^6 + 16 * q^7 + 8 * q^8 + 9 * q^9 $$q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{5} - 6 q^{6} + 16 q^{7} + 8 q^{8} + 9 q^{9} + 12 q^{10} - 12 q^{12} - 38 q^{13} + 32 q^{14} - 18 q^{15} + 16 q^{16} + 126 q^{17} + 18 q^{18} - 20 q^{19} + 24 q^{20} - 48 q^{21} + 168 q^{23} - 24 q^{24} - 89 q^{25} - 76 q^{26} - 27 q^{27} + 64 q^{28} - 30 q^{29} - 36 q^{30} - 88 q^{31} + 32 q^{32} + 252 q^{34} + 96 q^{35} + 36 q^{36} + 254 q^{37} - 40 q^{38} + 114 q^{39} + 48 q^{40} - 42 q^{41} - 96 q^{42} + 52 q^{43} + 54 q^{45} + 336 q^{46} - 96 q^{47} - 48 q^{48} - 87 q^{49} - 178 q^{50} - 378 q^{51} - 152 q^{52} + 198 q^{53} - 54 q^{54} + 128 q^{56} + 60 q^{57} - 60 q^{58} - 660 q^{59} - 72 q^{60} + 538 q^{61} - 176 q^{62} + 144 q^{63} + 64 q^{64} - 228 q^{65} + 884 q^{67} + 504 q^{68} - 504 q^{69} + 192 q^{70} + 792 q^{71} + 72 q^{72} - 218 q^{73} + 508 q^{74} + 267 q^{75} - 80 q^{76} + 228 q^{78} + 520 q^{79} + 96 q^{80} + 81 q^{81} - 84 q^{82} + 492 q^{83} - 192 q^{84} + 756 q^{85} + 104 q^{86} + 90 q^{87} + 810 q^{89} + 108 q^{90} - 608 q^{91} + 672 q^{92} + 264 q^{93} - 192 q^{94} - 120 q^{95} - 96 q^{96} + 1154 q^{97} - 174 q^{98}+O(q^{100})$$ q + 2 * q^2 - 3 * q^3 + 4 * q^4 + 6 * q^5 - 6 * q^6 + 16 * q^7 + 8 * q^8 + 9 * q^9 + 12 * q^10 - 12 * q^12 - 38 * q^13 + 32 * q^14 - 18 * q^15 + 16 * q^16 + 126 * q^17 + 18 * q^18 - 20 * q^19 + 24 * q^20 - 48 * q^21 + 168 * q^23 - 24 * q^24 - 89 * q^25 - 76 * q^26 - 27 * q^27 + 64 * q^28 - 30 * q^29 - 36 * q^30 - 88 * q^31 + 32 * q^32 + 252 * q^34 + 96 * q^35 + 36 * q^36 + 254 * q^37 - 40 * q^38 + 114 * q^39 + 48 * q^40 - 42 * q^41 - 96 * q^42 + 52 * q^43 + 54 * q^45 + 336 * q^46 - 96 * q^47 - 48 * q^48 - 87 * q^49 - 178 * q^50 - 378 * q^51 - 152 * q^52 + 198 * q^53 - 54 * q^54 + 128 * q^56 + 60 * q^57 - 60 * q^58 - 660 * q^59 - 72 * q^60 + 538 * q^61 - 176 * q^62 + 144 * q^63 + 64 * q^64 - 228 * q^65 + 884 * q^67 + 504 * q^68 - 504 * q^69 + 192 * q^70 + 792 * q^71 + 72 * q^72 - 218 * q^73 + 508 * q^74 + 267 * q^75 - 80 * q^76 + 228 * q^78 + 520 * q^79 + 96 * q^80 + 81 * q^81 - 84 * q^82 + 492 * q^83 - 192 * q^84 + 756 * q^85 + 104 * q^86 + 90 * q^87 + 810 * q^89 + 108 * q^90 - 608 * q^91 + 672 * q^92 + 264 * q^93 - 192 * q^94 - 120 * q^95 - 96 * q^96 + 1154 * q^97 - 174 * q^98

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
2.00000 −3.00000 4.00000 6.00000 −6.00000 16.0000 8.00000 9.00000 12.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.4.a.f 1
3.b odd 2 1 2178.4.a.e 1
11.b odd 2 1 6.4.a.a 1
33.d even 2 1 18.4.a.a 1
44.c even 2 1 48.4.a.c 1
55.d odd 2 1 150.4.a.i 1
55.e even 4 2 150.4.c.d 2
77.b even 2 1 294.4.a.e 1
77.h odd 6 2 294.4.e.h 2
77.i even 6 2 294.4.e.g 2
88.b odd 2 1 192.4.a.i 1
88.g even 2 1 192.4.a.c 1
99.g even 6 2 162.4.c.c 2
99.h odd 6 2 162.4.c.f 2
132.d odd 2 1 144.4.a.c 1
143.d odd 2 1 1014.4.a.g 1
143.g even 4 2 1014.4.b.d 2
165.d even 2 1 450.4.a.h 1
165.l odd 4 2 450.4.c.e 2
176.i even 4 2 768.4.d.c 2
176.l odd 4 2 768.4.d.n 2
187.b odd 2 1 1734.4.a.d 1
209.d even 2 1 2166.4.a.i 1
220.g even 2 1 1200.4.a.b 1
220.i odd 4 2 1200.4.f.j 2
231.h odd 2 1 882.4.a.n 1
231.k odd 6 2 882.4.g.f 2
231.l even 6 2 882.4.g.i 2
264.m even 2 1 576.4.a.q 1
264.p odd 2 1 576.4.a.r 1
308.g odd 2 1 2352.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 11.b odd 2 1
18.4.a.a 1 33.d even 2 1
48.4.a.c 1 44.c even 2 1
144.4.a.c 1 132.d odd 2 1
150.4.a.i 1 55.d odd 2 1
150.4.c.d 2 55.e even 4 2
162.4.c.c 2 99.g even 6 2
162.4.c.f 2 99.h odd 6 2
192.4.a.c 1 88.g even 2 1
192.4.a.i 1 88.b odd 2 1
294.4.a.e 1 77.b even 2 1
294.4.e.g 2 77.i even 6 2
294.4.e.h 2 77.h odd 6 2
450.4.a.h 1 165.d even 2 1
450.4.c.e 2 165.l odd 4 2
576.4.a.q 1 264.m even 2 1
576.4.a.r 1 264.p odd 2 1
726.4.a.f 1 1.a even 1 1 trivial
768.4.d.c 2 176.i even 4 2
768.4.d.n 2 176.l odd 4 2
882.4.a.n 1 231.h odd 2 1
882.4.g.f 2 231.k odd 6 2
882.4.g.i 2 231.l even 6 2
1014.4.a.g 1 143.d odd 2 1
1014.4.b.d 2 143.g even 4 2
1200.4.a.b 1 220.g even 2 1
1200.4.f.j 2 220.i odd 4 2
1734.4.a.d 1 187.b odd 2 1
2166.4.a.i 1 209.d even 2 1
2178.4.a.e 1 3.b odd 2 1
2352.4.a.e 1 308.g odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(726))$$:

 $$T_{5} - 6$$ T5 - 6 $$T_{7} - 16$$ T7 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 3$$
$5$ $$T - 6$$
$7$ $$T - 16$$
$11$ $$T$$
$13$ $$T + 38$$
$17$ $$T - 126$$
$19$ $$T + 20$$
$23$ $$T - 168$$
$29$ $$T + 30$$
$31$ $$T + 88$$
$37$ $$T - 254$$
$41$ $$T + 42$$
$43$ $$T - 52$$
$47$ $$T + 96$$
$53$ $$T - 198$$
$59$ $$T + 660$$
$61$ $$T - 538$$
$67$ $$T - 884$$
$71$ $$T - 792$$
$73$ $$T + 218$$
$79$ $$T - 520$$
$83$ $$T - 492$$
$89$ $$T - 810$$
$97$ $$T - 1154$$